SEC. I.j OF NATURAL PHILOSOPHY. 253 G, g ; and the body ascending will in the time DGgd describe the space EG«-e; in the time DGBA, the space of the whole ascent EGB ; in the time ABK1, the space of descent BFK ; and in the time IKki the space of descent KFfk; and the velocities of the bodies (proportional to the re sistance of the medium) in these periods of time will be ABED, ABed, O, ABFI, AB/z respectively ; and the greatest velocity which the body can acquire by descending will be BACH. For let the rectangle BACH be resolved into in numerable rectangles AA', K/, Lm, M//, *fea, which shall be as the increments of the velocities produced in so many equal times; then will 0, AAr, AL Am, An, ifec., be as the whole velocities, and therefore (by suppo sition) as the resistances of the medium in the be- ginning of each of the equal times. Make AC to AJLLB AK, or ABHC to AB/vK, as the force of gravity to the resistance in the beginning of the second time ; then from the force of gravity subduct the resistances, and ABHC, K/vHC, L/HC, MwHC, (fee., will be as the abso lute forces with which the body is acted upon in the beginning of each of the times, and therefore (by Law I) as the increments of the velocities, that is, as the rectangles AA-, K/, Lm, M//, (fee., and therefore (by Lem. 1, Book II) in a geometrical progression. Therefore, if the right lines K£, LI M/TO, N//, &c., are produced so as to meet the hyperbola in q, r, s, t, (fee.. the areas AB^K, K